Mathematical modeling and analysis of tumor-immune interactions 6 - - PowerPoint PPT Presentation

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

Mathematical modeling and analysis of tumor-immune interactions

Kevin Atsou 1 Thierry Goudon 1 Biologists : Véronique Braud 2 Fabienne Anjuere 2

1Université Côte d’Azur, Inria (Team COFFEE), CNRS, LJAD 2CNRS, IPMC (Institut de Pharmacologie Moléculaire et Cellulaire)

6 août 2019

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

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tumors and efgector T-cells interactions : Biological context

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Mathematical modeling : Earlier stages of tumor-growth

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Numerical simulations

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations The Immune system

Summary

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tumors and efgector T-cells interactions : Biological context

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Mathematical modeling : Earlier stages of tumor-growth

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Numerical simulations

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations The Immune system

The Immune system Attacks Tumor cells

Immunotherapy starts in 1891 with W. Coley (Coley’s toxins) ; Immunotherapy restarted seriously With the appearance of AIDS ; immunodefjcient patients

• ften develop cancer

Immune cells play an important role in the control

• r in the development of

tumors

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source : https ://gifer.com/en/7ftZ

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations The Immune system

Two types of Immune system

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source : https ://www.biosa.co.nz/, what is your immune system

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations The Immune system

Tumor cells cycle

Genes involved in cell cycle control are subject to the genetic alterations = ⇒ cancer = ⇒ Uncontrolled Cell division

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3.

source : https ://curioussciencewriters.org/

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations The Immune system

Tumor Immunity Cycle

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4.

source : http ://biocc.hrbmu.edu.cn/TIP/

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations The Immune system

K.Atsou - Impediments of the tumor immunity cycle

Tumor has many suppressive infmuences The prevalence of cancer indicates that the immune system does not have a strong enough efgect on Tumors. high levels of suppressive cytokines ; T regulatory cells (Tregs produce TGF-β and IL-10) ; high expression of PD-L1 (Programmed death-ligand 1 ) by tumors ; Production of VEGF (Vascular Endothelial Growth Factor) for angiogenesis ;

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

Summary

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tumors and efgector T-cells interactions : Biological context

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Mathematical modeling : Earlier stages of tumor-growth

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Numerical simulations

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

A growth-fragmentation modeling approach : Tumor

(t, z) → n(t, z) (in celln · µm−3) the size-structured tumor cells distribution (t, x) → c(t, x) (in cellc · mm−3) the concentration of Efgector T cells

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

Tumor growth main processes

Tumor growth is splitted into two main processes : natural (microscopic) growth of the cells, cell division.

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source : https ://gifer.com/en/9uMp

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

A growth-fragmentation modeling approach : Tumor

We model the tumor growth by the following equation : ∂ ∂tn(t, z) = − ∂ ∂z(V(z)n(t, z))

• the cell’s volume growth

+ Q(n(t, z))

• Cellular division operator

. We can complete the equation by the initial data, n(t = 0, z) = n0 and the boundary condition, n(t, 0) = 0

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

The cellular division

∂ ∂tn(t, z) = − ∂ ∂z(V(z)n(t, z))

• the cell’s volume growth

+ Q(n(t, z))

• Cellular division operator

. Q(n(t, z)) ? a(z), the rate at which cells of size z process division. k(z′|z) the distribution of products from a cells of size z dividing.

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

The cellular division

k(z′|z) must be normalized so that mass is conserved : ∫ z z′k(z′|z)dz′ = z. (1) Consequently, Q(n) = −a(z)n(t, z)

• The loss of cells of size z

+ ∫ ∞

z

a(z′)k(z|z′)n(t, z′)dz′

• gain of cells with size z

. (2)

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

Binary and symmetric The cellular division

Q(n) = −a(z)n(t, z)

• The loss of cells of size z

+ ∫ ∞

z

a(z′)k(z|z′)n(t, z′)dz′

• gain of cells with size z

. (3)

We assume binary and symmetric division process. Therefore cells of size 2z give birth to cells of size z. k(z|2z) = 2δz′=2z Q(n(t, z)) = −a(z)n(t, z) + 2a(2z)n(t, 2z)d(2z) (4) Q(n(t, z)) = 4a(2z)n(t, 2z)

• gain of cells with size z

− a(z)n(t, z)

• The loss of cells of size z

(5)

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

Chemotaxis phenomenon

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T cells displacement towards the tumor microenvironment is model by chemotaxis.

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

K.A - The antigen-specifjc CD8+ efgector T cells displacement

the activated T cells follows the gradient of the chemical signal Let’s denote by x

N x

R c t the time dependent concentration of immune cells in a volume : c t c t x dx (6)

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

K.A - The antigen-specifjc CD8+ efgector T cells displacement

the activated T cells follows the gradient of the chemical signal Let’s denote by Ω = {x ∈ RN, |x| ≤ R}, cω(t) the time dependent concentration of immune cells in a volume ω ⊂ Ω : cω(t) = ∫

ω

c(t, x)dx (6)

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

The tumor mass : kth order moments

Let’s denote by µ0(t) (the zeroth order moment) the total number of tumor cells in the tumor ; µ0(t) = ∫ ∞ n(t, z)dz (7) Let’s denote by µ1(t) (the fjrst order moment) the total volume of the tumor ; µ1(t) = ∫ ∞ zn(t, z)dz (8)

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

The Chemotaxis model

∂c ∂t(t, x)+

Convection term

• ∇ · (χΦ(∇xΦ)c) −

Difgusion term

D(∇xc) =

Conversion of Im. cells

• pfg(µ1)S

−

Natural Death

• γc

(9) c|∂Ω = 0 Typical conversion law of Naive T cells (Saturated conversion). g(µ1) = µ1 δ + µ1 (10)

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

Tumor antigen-induced potential

The attractive potential Φ satisfjes a Poisson equation : −∇ · (k(x)∇Φ) = ⟨σ⟩ ∫ ∞ zn(t, z)dz, −∇ · (k(x)∇Φ) = µ1(t)⟨σ⟩, With Neumann homogeneous boundary condition. k(x)∇xΦ(x) · n∂Ω = 0 where ⟨σ⟩ = σ −

1 |Ω|

∫

Ω σ dx

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

A chemotaxis model : Immune cells

The resulting Chemotaxis model                   

∂tc(t, x) + ∇ · (χ(φ)(∇xφ)c − (D(x)∇xc)) = pg(µ1)S − γc, ∀t > 0, x ∈ Ω −∇ · (k(x)∇φ) = ∫ ∞ zn(t, z)σ(x, z)dz, ∀t, z > 0, x ∈ Ω c = 0, ∀x ∈ ∂Ω, c(t = 0, x) = c0(x), ∀x ∈ Ω, k(x)∇φ · n∂Ω = 0 ∀x ∈ ∂Ω.

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

Tumor vs Immune cells : the interactions term

Let’s denote by m(c, n) the death term describing the immune response The tumor growth model becomes : ∂ ∂tn(t, z) = − ∂ ∂z(Vn(t, z)) + Q(n(t, z)) − m(c, n)

The interaction Tumor-Im. Cells

. m c n t y c t y dy n t z n t z where is an interaction kernel

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

Tumor vs Immune cells : the interactions term

Let’s denote by m(c, n) the death term describing the immune response The tumor growth model becomes : ∂ ∂tn(t, z) = − ∂ ∂z(Vn(t, z)) + Q(n(t, z)) − m(c, n)

The interaction Tumor-Im. Cells

. m(c, n)(t) = ∫

Ω

δ(y)c(t, y)dy × n(t, z) α + n(t, z). where δ is an interaction kernel

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

Asymptotic behavior : Tumor growth

We turn back to the tumor growth equation without immune response :       

∂tn(t, z) = −∂z(V(z)n(t, z)) − a(z)n(t, z) + 4a(z)n(t, 2z), ∀(t, z) ∈ R⋆

+ × R+

n(t, 0) = 0, ∀t ≥ 0, n(t = 0, z) = n0(z) ∈ L1(R+), ∀z ≥ 0,

The solution behaves like : (t, z) → ρeλtN(z) with ρ = ∫

R+ n0(z)

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

Asymptotic behavior : Tumor growth

The long time behavior of n is directly linked with the existence

• f a solution λ, N, φ, in L1 sense, of the associated

eigenproblem,             

∂z(V(z)N(z)) + (λ + a(z))N(z) = 4a(z)N(2z), z ≥ 0 N(0) = 0, N(z) > 0 for z > 0, ∫ ∞

0 N(Z)dz = 1

V(z)∂zψ(z) − (λ + a(z))ψ(z) = −2a(z)ψ(z 2), x ≥ 0 ψ(z) > 0 for x ≥ 0, ∫ ∞

0 N(z)ψ(z)dz = 1

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

Asymptotic behavior : Tumor growth

The solution of the dual equation allows to defjne a conservation law

• n n(t, z) :

Mass Conservation ∫ ∞ n(t, z)e−λtψ(z)dz = ∫ ∞ n0(z)ψ(z)dz

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

Existence and uniqueness of the eigenelements

Theorem, B. Perthame et al., 2005 if the division rate a is lower bounded and upper bounded, then there is a unique solution (λ, N, ψ) to the eigenproblem with ψ, N in C1(R+) and moreover, all the moments of the asymptotic state N are bounded. The existence of the eigenelements is based on the Krein-Rutman Theorem

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

Asymptotic behavior : Tumor growth

Exponential decay to the leading eigenpair of the growth-fragmentation operator : Theorem, B. Perthame et al., 2004 If the division rate and the growth rate of the tumor are constant, there is a unique solution to the fjrst eigenvalue

• problem. This solution belongs to the Schwartz space and

furthermore, all the solutions to the tumor growth equation converges exponentially to the fjrst eigenvector with a rate equal to the tumor division rate.

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

Asymptotic behavior : Tumor growth

There is a semi-explicit formula for the leading eigenfunction N (F. Baccelli, 2002, B. Perthame et al., 2004 ) given by a series that converges absolutely and uniformly Lemma, F.Baccelli, 2002 Let α0 = 1, αn = 2 2n − 1αn−1, then the function N(z) = ⟨N⟩

∞

∑

n=0

(−1)nαn exp ( −2n+1 a Vz ) , belongs to S(R+) and is unique.

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

Asymptotic behavior : Tumor growth

difgerent profjles :

0.5 1 1.5 2 2.5 z 0.5 1 1.5 2 2.5 3 3.5 4 Equilibrium solution a/V = 1 a/V = 1.2 a/V = 1.5 a/V = 2 a/V = 3 a/V = 3.5 a/V = 4

An example of time evolution :

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

Asymptotic behavior : Tumor growth

Existence and uniqueness of eigenelements has been proved Residual distribution the system reaches a certain equilibrium state, with a non zero tumor cells distribution In which cases the immune system can control the tumor progression ?

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

Asymptotic behavior : Immune cells taking control

An attempt which clarifjes this issue is provided by the following statement for small division rates (non-aggressive tumors) Proposition, Kevin ATSOU & Thierry Goudon If the tumor is non-aggressive, there exist a unique tumor mass which depends on the tumor division rate at which the tumor is always controlled by the immune cells.

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

Summary

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tumors and efgector T-cells interactions : Biological context

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Mathematical modeling : Earlier stages of tumor-growth

3

Numerical simulations

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

model parameters

Variable parameters :

(a) The T cells Strength δ (b) The chemical signal σf

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

Homogeneous distribution of Naive cells : a = 1

Tumor vs Homogeneous distribution of Immune cells :

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

Homogeneous distribution of Naive cells : a

(c) a = 0.0625 (d) a = 0.25 (e) a = 4 (f) a = 16 Figure – behavior of the solutions. (µ1 in red, µc in blue, a in black)

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

Heterogeneous distribution of Naive cells

Tumor vs Heterogeneous distribution of Immune cells :

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

Heterogeneous distribution of Naive cells : a = 4

Tumor vs Heterogeneous distribution of Immune cells :

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

Heterogeneous distribution of Naive cells

(a) a = 0.0625 (b) a = 0.25 (c) a = 4 (d) a = 16 Figure – Evolution of the tumor mass µ1 (red curves, left axis), and of ¯ µc (blue curve, right axis) for several values of the division rate a.

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019

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tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations

Thank you for your attention...

Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019